5/16
Figure 1-3-18. Configuration of sample electrode coating. Dimensions are in inches.
4. Sample Description
The sample is a single crystal stoichiometric MgA1 20
4 spinel grown by Czochra1ski method and furnished by Crystal Products Division of Union Carbide Corporation. After ~ough cut, the boule is lapped
into a prism with two sets of faces parallel to (001) and (110) respec- tively. The crystal orientation is detennined to be 5± 5' parallel to
(001) and 9± 5' parallel to (1l0) by Laue back reflection method. (The X-ray work is done by Dr. E. K. Graham, now at Pennsylvania State Uni- versity). The final polish of the two sets of faces to 1/10 wavelength
{sodium 1igh~f1at and laser finish is done by Crystal Optics (Ann Arbor, Mich.). The parallelism between the polished faces is less than 10 seconds of arc. The dimension of the sample is 0.8923± 0.0002 cm
(between (001) faces) x 1.1793± 0.0002 cm (between (1l0) faces) x 1. 2578r± :lhOOQ2 Cirl'16etweiHf'the-ulJpeti sbed.:(11 0) faces). The dens i ty of the sample is measured by immersion method using distilled water as
immersion liquid. At 180C the density is measured to be 3.5790 gm/cm3 . The density is computed to be 3.5784 gm/cm3 at 250C using
a
=
6.93 x 10- 6/K for linear thenna1 expansion coefficient (Rigby et a1 1946). The sample is also placed between two crossed po1arizers and illuminated by a diffused light source to examine for residual stress.No residual stress is found within the sample between the two sets of faces both before and after the experiment.
-43-
5. Data and Data Reduction
The data as recorded (raw data) are two dark spots exposed by the diffracted laser beam on a photographic plate. A contact print made from such a photographic plate is shown in Figure 1.5-1. The distance between the two spots is measured on the optical density readout recorder chart from a Joyce-Loeb1 double beam microdensitom- eter (model MkIIICS, serial no. 497, Joyce, Loeb1 and Company,
Burlington, Mass.). The details of the distance measurement are des- cribed below. The photographic plate is placed on the microdensitom- eter stage with its bottom edge pushed against the bottom edge of the recessed stage. The direction defined by the two spots on the photo- graphic plate is aligned parallel to the scanning direction of the microdensitometer by rotating the stage and observing the images of the two spots with respect to a cross hair on the microdensitometer viewing screen. Alignment error is ±0.1°. The angle between the direction defined by the spots and bottom edge of the photographic plate y is read off the stage rotation table. A Starrett no. 360 precision protractor (L. S. Starrett Company, Athol, Mass.) is then placed on top of the photographic plate with its square frame set against the edge of the microdensitometer stage. The protractor angle
is set at the angle y but in an opposite sense. The vernier on the Starrett no. 362 protractor reads to 1/12 of a degree. An Ameri can Optical Corporation A01400 stage micrometer (Scientific Instrument Division, American Optical Corporation, Buffalo, N.Y.) is laid on top of the two spots and against the blade of the precision protractor.
The positioning of the protractor angle puts the scale divisions on stage micrometer perpendicular to the microdensitometer travel direc- tion. The purpose of the stage micrometer is to put fiducial marks on the microdensitometer readout chart for distance measurement. The photographic plate, Starret protractor, and the stage micrometer are fixed in position on the microdensitometer stage by spring clips. An example of the microdensitometer readout with this arrangement is shown in Figure I.S-2a and Figure I.S-2b. Since it is the change of distance that is important in the temperature dependence measurement, the two spikes marked "4 rrm" and "14 rrm" are used as fixed fiducial marks in an entire suite of measurements. (For example, Vp[lOO] from 293 to 423K). The two density profiles are then matched on a light table and a pencil mark is put on both charts at the same {but arbi-
trar~ location. The distance between the spots is therefore equal to a fixed distance (distance between the fiducial marks "4 mm" and "14 mm") plus the distance from the pencil mark on Figure I.S-2a to "4 mm"
and the distance from the pencil mark on Figure I.S-2b to "14 mm".
The scale on the c~arts is provided by the distance between the fidu- cial marks "3 mm" and "4 rrm", and "14 mm" and "lS 1l111", respectively.
These distances are calibrated by American Optical Corporation to be 0.1 ± O.OOOS cm. The use of fiducial marks is necessitated by the limited travel of the microdensitometer stage at such a large magnifi- cation (approximately 200 to 1). The distance between the two spots in Figures I.S-2a and I.S-2b is then read as
-45-
( 9.23 ± 0.01 (6.98) ± 0.01
i= lOmn+ + 6.71
) mm 19.75 ± 0.01 19.81 ± 0.01
where 10 mn is the distance between the fixed fiducial marks "4 mn"
and "14 mn". 9.23 is the distance in centimeters on the chart from the pencil mark on Figure I.5-2b to the "14 mn" marker. 19.75 is the distance in centimeters on the chart between the fiducial marks
"14 mn" and "15 mn" on the same figure. The corresponding readings on Figure I.5-2a are (~:~~), and 19.81. There are two readings, 6.98 and 6.71 because there are two pencil marks on Figure I.5-2a. There are two pencil marks on Figure I.5-2a because the two density profiles are not identical, and there is a latitude in which "matching" of the
two density profiles can be considered as equally good. In the pres- ent case 6.98 is the distance in centimeters on the chart from the pencil mark to the "4 mm" marker when matching is by the top part of the density profiles. 6.71 is the corresponding distance when match- ing is by the bottom part of the density profiles. These are illus- trated in Figures 1.5-3 and 1.5-4. This latitude in matching is indeed the major source of experimental uncertainty, and will be dis- cussed in detail later. A measure of the reproducibility of the microdensitometer readout is provided by the repeated optical density readouts of divisions on the stage micrometer used as fiducial
markers. Histograms of the distribution of repeated readout of dis- tance between pairs of markers are shown in Figure 1.5-5. It is estimated from these histograms that the repeatability of intensity
profiles is within 0.05 cm on the microdensitometer recorder chart.
Note that in Figure 1.5-5 the distance between the two markers "5 mm"
and "6 mrn" in the data readout for Vs[OOl] differs from that in the data readout for Vp[llOJ. This is because the ratio arm connecting the microdensitometer specimen table and recorder chart table is re- moved between the two readouts, and slight variation in the setting of
the ratio arm in its socket causes the difference in the two readouts.
Since the error in aligning the two spots with the microdensitometer stage travel direction is ±O.lo, and the error in aligning the stage micrometer with the two spots is also ±O.lo, the error in distance measurement between the two spots due to angular misalignment is
t(l - cos 0.140) ~ 5 x 10-6t ~ 0.05 pm , equivalent to 0.001 cm on the readout chart, and is negligible compared to the latitude in densit-y
prof:~He~tching as discussed above.
Consider now the problem of the difference in the two density profiles on the same photographic plate. Note first that the degree of disparity varies from one photographic plate to another. Indeed, the two density profiles in Figures I.5-2a and I.5-2b are dne of the worst cases. An example of two density profiles which match better is illustrated by Figures I.5-6a, I.5-6b, and 1.5-7. Note secondly, that whenever "bad" matching exists, the matching of one profile with the mirror image of the other profile is generally better. This is illus- trated in Figure 1.5-8 with the profile in Figure I.5-2a matching the mirror image of the profile in Figure I.5-2b. There are density pro- file distortions introduced during photographic processing, such as unevenness in plate emulsion and water stain marks during drying, and electronic noises introduced during the microdensitometer readout.
-47-
These distortions and noises are, however, insufficient to account for the disparity, when it exists, between the two density profiles.
The reason for this disparity is that the stress-induced birefringence inside the crystal does not assume a plane wave front parallel to the sample faces. Temperature gradient inside the sample, edge and corner effects, lack of perfect parallelism between sample faces, positioning of the transducer (not exactly at the center of sample face) and
transducer radiation pattern all cause the steady state birefringence
pattern~to deviate from a plane standing wave. However, the steady state birefringence pattern induced by forced vibration of the crystal can be decomposed into plane waves in different directions.
These plane waves will be divided into those whose wave vectors are perpendicular to the light wave vector and those which do not belong to the class just described for separate discussions. For plane waves whose wave vector are perpendicular to the light wave vector,
the diffracted light intensity is given by equation (1.2-22) with a
=
0, i.e., 1 21
=
Jl(v). By adjusting the transducer driving fre- quency, however, only plane waves whose wave vector lies nearly per- peodtcular to the polished faces (on one of which the transducer is mounted) can enjoy the condition of constructive interference. The result is demonstrated in Fig. 1.5-1. Since all photographic plates are pushed with bottom edge against the film holder bottom orienting screws during exposure, the direction defined by the two spots rela- tive to the photographic bottom edge serves to check the maximum deviation in direction from pure mode of the sound wave velocitywhich each photographic plate measures. The microdensitometer table angle readings in Tables 1.5-1 to 1.5-4 are exactly these data.
Since to compute, from pure mode velocity, sound velocities of modes whose directions deviate slightly from a pure mode involves cor-
rection factors proportional to the square of the cosine of the deviation angle (for example, Neighbours and Schacher 1967) and
1 - cos 20.6° = 1.08 x 10-4, where 0.60 is the maximum deviation angle in Tables 1 through 4, the error involved in this respect is at most one-fifth of that arising from disparity of density profHes·,-. In the more typical case, the deviation angle is 0.20 and 1 - cos 20.2° = 1.2 x 10-5. Now consider those waves in whose wave front the light wave vector does not lie. The diffracted light intensity from these
( ) . _ 2[ sin(QaL2)]
waves is given by equation 1.2-22 wlth a
r
0, 11- J l v (Qa/2) . The intensity distributionJ~[v
slQ£?2)2)] as a function of a is plotted in Klein and Cook (1967) and has zeros atOn
= 2mm where m is any non-zero integer. The intensity is significant only for values of a less than the first zero. The first zero occurs at a = 2n/Q . Recall Q = 2n ~ A t , a =_n ~ sin e • where e is the angle betweenA2 no 0 AO
sound wave front and light wave vector.
On
= 2n implies Isin el= A/t.Take t to be width of the sample e ~ 1030' for Vp[lOO] measure- ments, 10 for Vs[lOO] measurements, 20 for Vp[llO] measurements, and 0050' for Vs[llO] measurements. Since the characteristic curve
(Hurter-Driffield curve) of the Agfa-Gavaert 8E75 and 10E75 plates shown in Figure 1.5-9 is such that exposure for energy density below the "toe" is greatly depressed, the effective angular range in which
-49-
light diffracted from these waves is recorded on the photographic plate is even less than the angles listed above. Nevertheless, this does have the effect of causing disparity between the two density pro- files and explains also why in case of disparity, the match between one density profile with mirror image of the other profile is better than the match between the two profiles themselves.
The data interpreted as described above are listed in Tables 1.5-1 through 1.5-4. Each table provides data for velocity measurement as a function of temperature of a pure mode. There are two spot
separation readings for each photographic plate. The first reading is the separation when matching the top part of the two profiles and the second reading is the separation when matching bottom part of the two profiles. The first reading is not consistently greater than the second, nor is it true vice versa. It depends rather on the detailed excitation of off-pure mode plane waves which in turn depends on exact sample geometry, positioning of transducer, transducer driving fre- quency, and temperature gradient inside the sample. In computing these distances, the distance between the fixed markers is taken as the cali- brated distance shown on the stage micrometer. Any error in this distance affects only the absolute velocity but not the temperature dependence of velocity measurements. In summary, the errors discussed in the measurement of the distance between the two spots on the photo- graphic plates by the microdensitometer readout are:
(1) Error due to m1crodensitometer and stage micrometer mis- alignment. This is about ~.l ~m;
(2) Pencil marker and recorder pen trace thickness on the
microdensitometer readout chart. This error is less than 1 ~;
(3) Reproducibility of microdensitometer readout, which is within 2.5 ~m.
(4) The error involved when the direction defined by the two spots, relative to the photographic plate bottom edge, fluc- tuates from plate to plate. This error is typically one order of magnitude smaller than and at most one-fifth of the uncertainty in profile matching.
(5) Error involved in profile matching, on the average~ 5 ~m.
Item (1) is negligible compared to item (5). Items (2) and (3) have a Gaussian probability distribution and their contribution to the stand- ard deviation of temperature dependence of velocity is negligible compared to item (5). Item (4) is one order of magnitude smaller than item (5), which leaves item (5) as the major contributing factor in experimental error.
Note also in Table 1.5-1,0-2,0-4,0-5,0-6, and 0-7 all meas- ure the same velocity at room temperature, only that the 0-5, 0-6, 0-7 measurements have one more half-wavelength inside the sample than the 0-2, 0-4 measurements. Also in Table 1.5-2, E-14, E-16, E-17 all measure the velocity at nearly the same temperature, but E-14 has a
slightly different frequency than E-16 and E-17. This is also the case in Table 1.5-3 between F-l, F-2, and F-6. These variations are used to check if this method would depend on the number of half wavelengths inside the sample, or the selection of transducer driving frequency.
From the readings of the ratio of transducer driving frequency and spot
-51-
separation as listed in Tables 1.5-1 to 1.5-4, these factors do not show any effect within the experimental error. It should also be noted that the effect of random index of refraction fluctuation along the partially evacuated light path has not been taken into account in the data analysis. Figures 1.5-10 to 1.5-13 are plots of the ratio of transducer driving frequency and spot separation vs. temperature con- structed from Tables 1.5-1 to 1.5-4, respectively. In these figures the short end of each error bar indicates density profile matching by top, while the long end indicates density profile matching by bottom.
A linear temperature dependence is fit to the quantity f/~
in Figures 1.5-10 through 1.5-13 by a least square method. The method is to minimize the sum of the distance from center of each error bar to the straight line squared. The intercept at OOC and slope of each least square fit straight line are labeled in each figure.
The standard deviation of the parameters of any least square fit scheme depends on the meaning of error bar at each individual data point. For example, if the error bar represents one standard devia- tion of a Gaussian distribution from repeated measurements at the same given value of the independent variable, the standard deviation of the parameters in the least square fit can be calculated, for example, according to Mathews and Walker (1965a). The error bars in the present case, as explained earlier, however, represent the latitude in the matching of the two density profiles, and the probability distribution
inside the error bars is not necessarily Gaussian. The assignment of standard deviation to the parameters is therefore difficult. Instead,
a measure of uncertainty in these parameters is assigned by a consis- tency check which will be discussed later.
Before computing the velocity and its temperature dependence from these results, two more possible sources of error must be consid- ered. The first is error in temperature measurement. As stated in the Experimental Techniques section, the temperatures are measured at two sites located at diagonally opposite corners of the sample. Each temperature reading listed in Tables 1.5-1 through 1.5-4 is an average of the temperatures at the two thermocouple sites. At room temperature the two thermocouples give identical readings. At higher temperatures the two readings differ. This difference is ~50C at 150oC. The cause for this difference is that the center of the sample holder assembly deviates slightly from the furnace center. This means that the temper- ature reading is uncertain to ±2.50C at 150oC. Since room temperature
is ~20oC for all measurements, this would introduce an error of
±2.50/130o
=
±2% to the slopes in Figures 1.5-10 through 1.5-13.Another possible source of error comes from the lensing effect at the exit sample face if a radial temperature gradient exists in the furnace.
A separate test is conducted to determine the upper limit of this radial thermal gradient. A fused silica 1 cm x 1 cm x 1 cm is placed inside the furnace in place of the spinel sample. A third thermocouple is pasted at tee center of the exit face (face toward the shutter) by Sauereisen cement. At ~lOOoC, the three thermocouple readings are 1.466 mV (thermocouple at one corner of the exit face), 1.660 mV
(thermocouple at the diagonally opposite corner) and 1.256 mV
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(thermocouple at center of the exit face). Since 0.1 mV corresponds to 2.220e and since the third thermocouple, placed inside the tubular furnace, provides a conduction path between high temperature region of the furnace and low temperature region outside the furnace (which is absent during the optical measurement), the maximum radial temperature difference existing in the optical measurement is estimated to be
o 3.25 0c 1 0e
4.44 e x 22.65 = 0.63 at '\, 00 . 3.25/22.65 is the ratio of tbermal conductivity of fused silica and spinel (Horai 1971). The lensing effect of the exit face can be estimated as the following.
Refer to Figure 1.5-14, x
=
6.93 x 10-6 ~T, where 6.93 x 10-6/ K is the linear thermal expansion coefficient of spinel (Rigby et al 1946), b • 0.8 cm is the distance between corner thermocouple and center thermocouple. The curvature a is given by a=
b2/2x=
0.64/{2 x 6.93 x 10-6 ~T). The change in diffraction angle due to lensing effect is then (for example, Yariv 1971a) ~e=
(nSPinel - 1)f =
0.719f =
15.6x
·10-6r~r,. Tbe corresponding change in spot separation on the photographicplate is ~i
=
L~e=
15.6 x 10-6 r~TL ,where L is the distance be- tween sample and photographic plate. Take Vs[OOlJ measurement, forA -4 6
example, r
=
'Ao x 1.179 cm=
0.6328 x 10 x 30.85 x 10 x 1.179 cm=
6.57 x 105
3.50 x 10-3cm , L
=
257.7 cm, ~i=
1.41 x 10-5 ~T cm/oe, i=
1.54 cm,~t/i
=
0.91 x 10-5 ~T. With d~~>~~/dT=
0.22 x 10-4 °e, the lensing effect introduces an error of 0.91 x 10 -5 x 0.63=
0.3% between room0.22 x 10-4 80
temperature and 1000e. This is neglected in comparison with other sources of error.
The velocity and its temperature dependence are computed from the values fit by the equations
t f
v = fA = (L + 2n . ) no (r)
splnel where t is the sample thickness;
and
AO is the laser wavelength in vacuum, AO
=
0.6328 ~m,the stability of He-Ne wavelength is within ~Ao/Ao =
~f/f
=
1.5 x 10 9=
3.16 x 10-6 , ~f being He-Ne laser 4.74 x 1014Dopp,ter wi dth.
~TV)P = (L + t ) 2A a{f~t))
a 2nspinel 0
a
p{I. 5-1)
(1. 5-2)
The values of velocity of the four pure modes at 250C and their tem- perature derivatives, together with values of Land t are listed in Table 1.5-5.
The four pure mode velocities are given by
(1.5-3)
where s s s
Cll' C12 and C44 are the three adiabatic elastic constants of spinel and p its density.